{
  "abstract": "We study second- and higher-order methods for solving smooth monotone variational inequalities (MVI). Monteiro and Svaiter (SIAM J. Optim., 2012) showed that a second-order method, NPE, converges at a rate of $\\mathcal{O}(T^{-1.5})$. For convex-concave minimax optimization, a subclass of MVI problems, Chen, Liu, Luo, and Zhang (COLT 2025) recently improved this rate to $\\tilde{\\mathcal{O}}( T^{-1.75})$. However, the result has a substantial gap compared to the lower bound of $Ω(T^{-2.5})$ established by Chen et al. (2026). In this paper, we propose a novel second-order method that achieves the optimal rate of $\\mathcal{O}(T^{-2.5})$. Our algorithm also extends to higher-order methods: for any integer $ p \\ge 1$, we obtain a $p$th-order method with a convergence rate of $\\mathcal{O}(T^{-(3p-1)/2})$, matching the known lower bounds and therefore establishing optimal rates across all orders.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.23557",
  "author_total": 6,
  "authors": [
    {
      "name": "Xinliang Zhang"
    },
    {
      "name": "Lesi Chen"
    },
    {
      "name": "Linxuan Pan"
    },
    {
      "name": "Chengchang Liu"
    },
    {
      "name": "Junchi Yang"
    },
    {
      "name": "Jingzhao Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/09qqp0z99nvy1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.23557",
  "fulltext_url": null,
  "id": "09qqp0z99nvy1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.23557",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.GT"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "openalex",
      "1703|Optimization and Variational Analysis"
    ]
  ],
  "openalex": "W7214040629",
  "pdf_url": "https://arxiv.org/pdf/2609.23557",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/09qqp0z99nvy1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.23557",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.23557"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "Optimal High-Order Methods for Solving Monotone Variational Inequalities",
  "title_text": "Optimal High-Order Methods for Solving Monotone Variational Inequalities",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:33Z",
  "url": "https://research.draconitic.com/paper/09qqp0z99nvy1",
  "venue": "arXiv",
  "versions": []
}